I. Teaching objectives 1. Knowledge and ability goal ① Make students understand the concept and descriptive definition of sequence limit. ② Make students judge the limits of some simple series, understand the "e-N" definition of series limits, and prove the limits of some series through step-by-step analysis.

(3) By observing the process of movement change, summarize the specific relationship between sequence and its limit, and improve students' mathematical generalization ability and abstract thinking ability.

2. The goal of process and method is to cultivate students' extreme thinking methods and autonomous learning ability.

3. Emotion, attitude and values make students understand the dialectical relationship between finiteness and infinity, approximation and precision, quantitative change and qualitative change, and cultivate students' dialectical materialistic views.

Second, the teaching focus and difficulties

Teaching emphasis: the concept and definition of sequence limit. Teaching difficulty: understanding the definition of sequence limit ε-n.

Third, the analysis of teaching objects

This lesson is the first lesson of sequence limit, and sequence limit is an introductory lesson for students to learn limit. For students, it is a brand-new content. Students' thinking is in the transitional stage from empirical abstract thinking to theoretical abstract thinking. When calculating the surface area and volume of a ball in the content of solid geometry, students have been exposed to extreme thinking, while in previous mathematics learning, they were mainly exposed to the problem of "limit" and rarely involved the problem of "infinity". The abstract concept of limit can make them intuitively understand and guide them to make a descriptive definition: "When n increases infinitely, the term an in the sequence {an} approaches the constant a infinitely, that is, the absolute value of the difference between an and a approaches zero infinitely", which can be used to judge the limit of some simple sequences. But mastering the "ε-N" language in one class is too demanding for them. Therefore, it is not appropriate to talk too hard, and we can study the limits of some simple sequences through specific examples. Make students understand the basic concept of limit and know what is the limit of sequence and its definition.

Fourthly, the design of teaching strategies and teaching methods.

This course adopts heuristic teaching method, and adopts multimedia courseware demonstration and student discussion to teach. Starting with a practical problem that students are familiar with, it can attract students' attention and stimulate their interest in learning. Then, through two simple series, the process that each item in the series approaches a constant infinitely with the increase of the number of items is demonstrated by multimedia courseware, so that students can discuss and summarize the characteristics of these two series on the basis of observation, and thus get a descriptive definition of the limit of the series. Then, under the guidance of the teacher, analyze the various situations of the limit of the sequence. So as to have an intuitive understanding of the limit of the sequence, and then let students judge the limit of some simple sequences according to the conditions of each item in the sequence. So as to achieve the effect of deepening the definition. Finally, consolidation exercises. Through such a complete teaching process, from observation to analysis, from quantitative to qualitative, from intuitive to abstract, with the help of multimedia courseware demonstration, students can gradually understand the new concept of limit, prepare for the operation and application of limit in the next class, and lay the foundation for learning advanced mathematics knowledge in the future. In the whole teaching process, pay attention to highlight the key points, break through the difficulties and achieve the requirements of teaching objectives.

Teaching process of verbs (abbreviation of verb)

1. Create a situation

Courseware shows creating situational animation.

Today we are going to learn a very important new knowledge.

Situation 1. Liu Hui, an ancient mathematician in China, founded "circumcision" in AD 263, saying that "cutting is fine and the loss is small. If you cut it, you cut it. You can't cut it. If you can encircle it, you can encircle it. There is no loss. "

Scenario 2: Zhuangzi written by Zhuang Zhou, an ancient philosopher in China? There is a quote in "Life on Earth": A foot pestle, half a day, lasts forever. That is to say, take a stick, cut it in half, take half and cut it in half to get a quarter, and then cut it in half to get an eighth? If we go on like this, we'll cut it indefinitely, half at a time. Will it be over?

As we all know, it is impossible to finish cutting, but every time you finish cutting, the stick is half less than the original one, which means that the length of the stick is getting shorter and shorter, but it will never become zero. This leads to the concept of limit.

2. Explore definitions

Explore the definition of the exhibition (1) animation demonstration.

Question 1: Please observe the following infinite series. When n increases infinitely, what are the characteristics of the changing trend of A and I?

( 1) 1/2，2/3，3/4，… n/n- 1 (2) 0.9，0.99，0.999，0.9999， 1- 1/0。

Teachers and students come to the following conclusions: When the number of terms n in the sequence (1) increases infinitely, the number of terms approaches to1infinitely; When the number n of terms in the sequence (2) increases infinitely, the number of terms is infinitely close to 1.

Then 1 is called the limit of sequence (1), and 1 is called the limit of sequence (2). The two series are only different in form. Both of them increase with the infinite number of terms n, and the number of terms approaches a constant infinitely, which is called the limit of this series.

So, what is the limit of the sequence? For an infinite sequence an, if an infinitely tends to a constant a when n increases infinitely, it is said that a is the limit of the sequence an.

Question 3: How to describe it quantitatively in mathematical language? How to describe the changing trend of the above series in mathematical language?

Show Definition Exploration (2) Animation demonstration, teachers and students * * * found that the smaller the distance between two points on the number axis, the closer the term is to 1, so we can use the infinitesimal distance between two points to describe the infinite approximation constant of the term. No matter how small a positive number E is specified in advance, if e=O- 1 is taken, an am can always be found in the sequence, so that the absolute value of the difference between all items after an item and 1 is less than ε, if E = 0. 000 1, then the absolute value of the difference between all items after item 6 and 1 is less than ε, that is, 1 is the limit of the sequence (1). Finally, teachers and students * * * summed up what quantities should be included in the definition of the limit of the sequence (these quantities are used to describe the limit of the sequence 1).

The limit of the sequence is: for any ε >; 0, if there is always a natural number n, when n >; The limit of inequality | an-a | nWhen n. Definition and exploration animation (1): Courseware can input any value n, calculate the value of the corresponding number n, and demonstrate the changing process of the sequence through animation. As shown in figure 1, it is the picture when the courseware is running. Define exploration animation (2) Courseware can input any value n, calculate the value corresponding to the nth item in the series and the value of an- 1I, and demonstrate the distance between the nth item and 1 with animation. Fig. 2 is a picture when the courseware is running.

3. Knowledge application

Here are three examples. Think with the students and analyze the answers together.

Example 1. Known series:

1，- 1/2， 1/3，- 1/4， 1/5……，(- 1)n+ 1 1/n，…

(1) Calculate |an-0|(2) The absolute value of the difference between all items after the first item and 0 is less than 0.0 17 and less than any specified positive number.

(3) Determine the limit of this series.

Example 2. Known series:

Known series: 3/2, 9/4, 15/8 ..., 2+(- 1/2)n, ...

Guess if this series has a limit, and if so, what should it be? It is found that the difference between each term and this limit is less than 0. 1 from the first term, and the difference between each term and this limit is less than 0.0 17 from the first term.

Example 3. Find the limit of constant sequence-7,7,7,7, ...

5. Summary of knowledge

In this lesson, we learned the concept of sequence limit and got a preliminary understanding of sequence limit. The limit of series studies the trend of infinite change. Through the discussion of the definition of series limit, we can see that this process is grasped through finiteness, and the dialectical relationship between finiteness and infinity, approximation and precision, quantitative change and qualitative change is fully reflected here.

After-class exercises:

(1) Judge whether the following sequence has a limit, and if so, find its limit value. ①an = 4n+l/n； ②an = 4-( 1/3)m； ③an =(- 1)n/3n； ④aan =-2； ⑤an = n； ⑥an=(- 1)n .

(2) Textbook exercise 1, 2.

Step 6 ask questions

Reflections on designing research-based learning.

Ask questions:

Zeno Paradox: Achilles is a running hero in Homer's epic. Running Achilles can never surpass the tortoise crawling slowly in front of him, because when Achilles reached the starting point of the tortoise, the tortoise had already walked a short distance ahead, Achilles had to catch up with this short distance again, and the tortoise moved forward again. In this way, Achilles can approach it infinitely, but he can't catch up with it. Suppose Achilles runs at a speed 10 times that of the tortoise, and the distance between Achilles and the tortoise is 1 km. If the tortoise runs 0. 1 km first, when Achilles catches up with 0. 1 km, the tortoise runs 0.05438+0 km forward. When Achilles caught up with 0.0 1 km, the tortoise ran forward by 0.00 1 km ... Can Achilles catch up with the tortoise?

The following is the content of research-based learning, taking the paradox that students are interested in as homework, consolidating the content learned in this section and further improving students' interest in learning the limit of sequence. At the same time, it also creates a situation for students to exchange and discuss after class, and gradually cultivates the habit of mutual cooperation, exchange and discussion, so that students can feel that mathematics comes from life and serves the essence of life, and gradually develop the habit of solving practical problems in life with mathematical knowledge.